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In his 1989 paper, Floer established a connection between holomorphic stripswith boundary on a Lagrangian $L$ and a small Hamiltonian push-off $L_{f}$, andgradient flow lines for the function $f$. The present paper studies thecompactness theory for holomorphic curves $u_{n}$ whose boundary components lieon Hamiltonian perturbations $L_{n}^{1},\dots,L^{N}_{n}$ of a fixed Lagrangian$L$, where each sequence of nearby Lagrangians $L^{j}_{n}$ converges to $L$ as$n\to\infty$. Generalizing earlier work of Oh, Fukaya, Ekholm, and Zhu, weprove that the limit of a sequence of such holomorphic maps is a configurationconsisting of holomorphic curves with boundary on $L$ joined by gradient flowlines connecting points on the boundary of holomorphic pieces. The key newresult is an exponential estimate analyzing the interface between theholomorphic parts and the gradient flow line parts.

Adiabatic compactness for holomorphic curves with boundary on nearby Lagrangians